How Do Gradients, Rates, and the Term 'Per' Relate to Division and Ratios?

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  • Thread starter Thread starter Miraj Kayastha
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Discussion Overview

The discussion explores the relationship between the term "per," gradients, rates, and their connection to division and ratios. It addresses conceptual understandings of speed, slope, and the mathematical representation of these ideas.

Discussion Character

  • Conceptual clarification, Technical explanation, Debate/contested

Main Points Raised

  • Some participants suggest that "per" in expressions like "3 miles per hour" implies division, interpreting it as "for each" hour, leading to the calculation of speed.
  • Others reference the equation y = mx + b, indicating that the slope (m) represents a ratio of the ordinate (y) to the abscissa (x), which can be interpreted as miles per hour when plotted with miles on the y-axis and time on the x-axis.
  • One participant introduces the concept of similar triangles, explaining that the tangent of an angle (Tan{Θ}) relates to the ratio of the lengths of the opposite and adjacent sides, suggesting that comparing these ratios is more meaningful than considering their absolute sizes.

Areas of Agreement / Disagreement

Participants express various interpretations of "per" and its implications for division and ratios, indicating that multiple competing views remain without a consensus on the most accurate explanation.

Contextual Notes

Some assumptions about the definitions of terms like "gradient" and "rate" are not explicitly stated, and the discussion does not resolve the mathematical implications of these concepts.

Miraj Kayastha
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Why does "per" in 3 miles per hour mean division?
Why are gradients and rates a ratio?
 
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Miraj Kayastha said:
Why does "per" in 3 miles per hour mean division?

The meaning of "per" is "for each" .

Now read 3 miles "for each" hour.
And you divide the 3 miles by one hour you will get the speed of an object.
 
Miraj Kayastha said:
Why does "per" in 3 miles per hour mean division?
Why are gradients and rates a ratio?

Are you not familiar with the basic equation
y = mx + b
where m is the slope, or rate ( and b is the intercept )

m is the ratio of the ordinate to the abscissa for any point on the line (x,y)

If you plot y-axis as the "miles" and x-axis as the time of hours , then the slope naturally follows as miles/hour, or in English terms miles per hour.

Same thing for gradient - for a surface that has a slope, its elevation will increase y-amount for every x-amount distance.
 
Why are gradients and rates a ratio?

Two "similar triangles" have the same angles but can be different sizes.

In each case

Tan{Θ} = Length of opposite side/length of adjacent side

So if interested in the angle or gradiant it makes sense to compare the ratio of the sides rather than their absolute magnitude.
 

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